1. Entropy Rate and Profitability
of Technical Analysis:
Experiments on the
NYSE US 100 Stocks
Nicolas NAVET – INRIA
France
nnavet@loria.fr
Shu-Heng CHEN – AIECON/NCCU
Shu-
Taiwan
chchen@nccu.edu.tw
CIEF2007 - 07/22/2007
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Outline
Entropy Rate : uncertainty remaining in the
next information produced given knowledge of
the past measure of predictability
Questions :
Do stocks exhibit differing entropy rates?
Does low entropy imply profitability of TA?
Methodology :
NYSE US 100 Stocks – daily data – 2000-2006
2000-
TA rules induced using Genetic Programming
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2. Estimating entropy
Active field of research in neuroscience
Maximum-likehood (“Plug-in”):
construct an n-th order Markov chain
not suited to capture long/medium term
dependencies
Compression-based techniques :
Lempel-Ziv algorithm, Context-Tree Weighting
Lempel- Context-
Fast convergence rate – suited to long/medium
term dependencies
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Selected estimator
Kontoyannis et al 1998
à n
!−1
ˆ 1X
hSM = Λi log2 n
n i=1
Λi : length of the shortest string that does
not appear in the i previous symbols
Example: 0 1 1 0 0 1 0 1 1 0 0
Λ6 = 3
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3. Performance of the estimator
Experiments :
Uniformly distributed r.v. in {1,2,..,8} –
r.v.
P
theoretical entropy = − 8 1/8 log2 p = 3 b.p.c.
i=1
Boost C++ random generator
Sample of size 10000
ˆ
hSM = 2.96
ˆ
Note 1 : with sample of size 100000, hSM ≥ 2.99
Note 2 : with standard C rand() function and
ˆ
sample size = 10000, hSM = 2.77
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Preprocessing the data (1/2)
pt
Log ratio between closing prices: rt = ln( pt−1 )
Discretization : {rt } ∈ R → {At } ∈ N
3,4,1,0,2,6,2,…
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3

4. Preprocessing the data (2/2)
Discretization is tricky – 2 problems:
How many bins? (size of the alphabet)
How many values in each bin?
Guidelines : maximize entropy with a number of
bins in link with the sample size
Here :
alphabet of size 8
same number of values in each bin
(“homogenous partitioning”)
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Entropy of NYSE US 100 stocks –
period 2000-2006
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20
Mean = Median = 2.75
Max = 2.79
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Min = 2.68
Density
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Rand() C lib = 2.77 !
5
0
2.66 2.68 2.70 2.72 2.74 2.76 2.78 2.80
entropy
Note : a normal distribution of same mean and standard deviation is
plotted for comparison.
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4

5. Entropy is high but price time
series are not random!
50
50
40
40
30
30
Density
Density
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20
10
10
0
0
2.65 2.70 2.75 2.80 2.85 2.65 2.70 2.75 2.80 2.85
Entropy (original data) Entropy (shuffled data)
Original time series Randomly shuffled
time series
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Stocks under study
Symbol Entropy
OXY 2.789
Highest entropy VLO 2.787
MRO 2.785
time series
BAX 2.78
WAG 2.776
Symbol Entropy
TWX 2.677
Lowest entropy EMC 2.694
C 2.712
time series JPM 2.716
GE 2.723
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5

6. Autocorrelation analysis
Typical high Typical low
complexity stock (OXY) complexity stock (EMC)
Up to a lag 100, there are on average 6 autocorrelations outside
the 99% confidence bands for the lowest entropy stocks versus 2
for the highest entropy stocks
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Part 2 : does low entropy
imply better profitability
of TA?
Addressed here: are GP-induced
GP-
rules more efficient on low-entropy
low-
stocks ?
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7. GP : the big picture
Training Validation Out-of-sample
interval interval interval
1 ) Creation of Further Performance
the trading rules selection on evaluation
using GP unseen data
2) Selection of -
the best
resulting One strategy is
strategies chosen for
out-of-sample
2000 2002 2004 2006
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GP performance assessment
Buy and Hold is not a good benchmark
GP is compared with lottery trading (LT) of
same frequency : avg nb of transactions
same intensity : time during which a position is held
Implementation of LT: random sequences
with the right characteristics, e.g:
e.g:
0,0,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0,1,1,1,1,1,1,…
0,0,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0,1,1,1,1,1,1,…
GP>LT ? LT>GP ? Student’s t-test at 95%
t-
confidence level – 20 GP runs / 1000 LT runs
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8. Experimental setup
Data preprocessed with 100-days MA
Trading systems:
Entry (long): GP induced rule with a classical
set of functions / terminals
Exit:
Stop loss : 5%
Profit target : 10%
90-days stop
90-
Fitness: net return - Initial equity: 100K$
- position sizing : 100%
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Results: high entropy stocks
GP net profits LT net profits GP>LT? LT>GP?
OXY 15.5K$ 14K$ No No
V LO 7K$ 11.5K$ No No
M RO 15K$ 18.5K$ No No
BAX 24K$ 13K$ Yes No
W AG 6K$ −0.5K$ Yes No
GP is always profitable
LT is never better than GP (at a 95% confidence
level)
GP outperforms LT 2 times out of 5 (at a 95%
confidence level)
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9. Results: low entropy stocks
GP net profits LT net profits GP>LT? LT>GP?
TWX −9K$ −1.5K$ No Yes
EM C −16.5K$ −11K$ No Yes
C 15K$ 18.5K$ No No
JP M 6K$ 10K$ No No
GE −0.5K$ 0.5K$ No No
GP is never better than LT (at a 95% confidence
level)
LT outperforms GP 2 times out of 5 (at a 95%
confidence level)
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Explanations (1/2)
GP is not good when training period is very
different from out-of-sample e.g.
out- of-
Typical low Typical high
complexity stock (EMC) complexity stock (MRO)
2000 2006 2000 2006
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10. Explanations (2/2)
The 2 cases where GP outperforms LT : training
quite similar to out-of-sample
out- of-
BAX WAG
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Conclusions
EOD NYSE time series have high but
differing entropies
There are (weak) temporal dependencies
Here, more predictable ≠ less risks
GP works well if training is similar to out-
of-sample
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11. Perspectives
Higher predictability level can be observed
at intraday timeframe (what about higher
timeframes?)
Experiments needed with stocks less
similar than the ones from NYSE US 100
Predictability tells us about the existence
of temporal patterns – but how easy /
difficult to discover them ??
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?
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